Any sequence of numbers with the property that any two consecutive numbers in the sequence are separated by a common difference.

tn=t1+(n-1)d

The sum of an arithmetic progression

Sn=(n/2)(t1+tn)

Geometric progression

A sequence of numbers with the property that the ratio of any two consecutive numbers in the sequence is constant.

tn=t1rn-1

The sum of a geometric progression

Sn=t1((1-rn)/(1-r))

Sn=t1((rn-1)/(r-1))

Sum of the infinite geometric progression

S=t1/(1-r)

We choose a focal date equal to one period before the first payment

The discounted value A of an ordinary simple annuity is defined as the equivalent dated value of the set of payments due at the beginning of the term.

A=

R (1-(1+i)-n)/i

R a n¬i

Discount factor for n payments / Discounted value of $1 per period.

R=A/an¬i=A/(1-(1+i)-n)/i

Annuity due

An annuity whose period payments are due at the beginning of each payment interval. The term starts at the time of the first payment and ends one payment period after the date of the last payment.

S=Rsn¬i(1+i)

A=Ran¬i(1+i)

Deferred annuity

An annuity with its first payment due some time after the end of the first interest period

A=Ran¬i(1+i)-k

The period of defferment is equal to the time of the first payment minus 1.

Forborne annuities

Has a period of time after the last payment of deposits is made and before the time when the accumulated value is calculated.

S=Rsn¬i(1+i)m

Balloon payment

The last regular payment is increased by a sum that will make the payment equivalent to the accumulated value S or the discounted value A.

Drop payment

A smaller concluding payment is made one period after the last full payment. Sometimes, when a certain sum of money is to be accumulated, a smaller concluding payment will not be required because the interest after the last full payment will equal or exceed the balance needed.

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General annuity

An annuity for which payments are made more or less frequently than interest is compounded.

Replace the given interest rate

A way to solve a general annuity problem. Do it by an equivalent rate for which the interest compounding period is the same as the payment period.

Replace the given payments

A way of solving a general annuity problem. Do it by equivalent payments made on the stated interest conversion dates.

Perpetuity

An annuity whose payments begin on a fixed date and continue forever.

Ordinary simple perpetuity

A=R/i

Simple perpetuity due

A=R/i + R

Ordinary simple perpetuity deferred k periods

A= R/i (1+i)-k

General perpetuity

An annuity when the payment interval and the interest period do not coincide with each other.

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